MA Day 51 – March 26, 2013 Section 13.1: (finish) Vector Fields Section 13.2: Line Integrals

Chapter 13: Vector Calculus

“In this chapter we study the calculus of vector fields, …and line integrals of vector fields (work), …and the theorems of Stokes and Gauss, …and more”

Section 13.1: Vector Fields

Wind velocity vector field 2/20/2007

Section 13.1: Vector Fields Wind velocity vector field 2/20/2007 Wind velocity vector field 2/21/2007

Section 13.1: Vector Fields Ocean currents off Nova Scotia

Section 13.1: Vector Fields Airflow over an inclined airfoil.

General form of a 2-dimensional vector field

Examples:

General form of a 2-dimensional vector field Examples: QUESTION: How can we visualize 2-dimensional vector fields?

General form of a 2-dimensional vector field Examples: Question: How can we visualize 2- dimensional vector fields? Answer: Draw a few representative vectors.

Example:

We will turn over sketching vector fields in 3- space to MAPLE.

Gradient, or conservative, vector fields

EXAMPLES:

Gradient, or conservative, vector fields EXAMPLES:

QUESTION: Why are conservative vector fields important?

ANSWER: Because when combined with F = ma, leads to the law of conservation of total energy. ( in section 13.3)

QUESTION: Why are conservative vector fields important? ANSWER: Because when combined with F = ma, leads to the law of conservation of total energy. ( in section 13.3) Sections 13.2 and 13.3 are concerned with the following questions:

QUESTION: Why are conservative vector fields important? ANSWER: Because when combined with F = ma, leads to the law of conservation of total energy. ( in section 13.3) Sections 13.2 and 13.3 are concerned with the following questions: 1.Given an arbitrary vector field, find a TEST to apply to the vector field to determine if it is conservative.

QUESTION: Why are conservative vector fields important? ANSWER: Because when combined with F = ma, leads to the law of conservation of total energy. ( in section 13.3) Sections 13.2 and 13.3 are concerned with the following questions: 1.Given an arbitrary vector field, find a TEST to apply to the vector field to determine if it is conservative. 2.Once you know you have a conservative vector field, “Integrate it” to find its potential functions.

Format of chapter 13: 1.Sections 13.2, conservative vector fields 2.Sections 13.4 – 13.8 – general vector fields

Section 13.2: Line integrals

GOAL: To generalize the Riemann Integral of f(x) along a line to an integral of f(x,y,z) along a curve in space.

We partition the curve into n pieces:

Assuming f(x,y) is continuous, we evaluate it at sample points, multiply by the arc length of each subarc, and form the following sum:

which is similar to a Riemann sum.

Assuming f(x,y) is continuous, we evaluate it at sample points, multiply by the arc length of each subarc, and form the following sum: which is similar to a Riemann sum.

EXAMPLE:

Extension to 3-dimensional space

Shorthand notation

Extension to 3-dimensional space Shorthand notation

Extension to 3-dimensional space Shorthand notation

Extension to 3-dimensional space Shorthand notation 3. Then

Line Integrals along piecewise differentiable curves